Derivation of the Polyakov action

We develop another method to get the Polyakov action, that is, the solution of the conformal Ward identity on a Riemann surface Σ. We find that this action is the sum of two terms: the first one is expressed in terms of the projective connection and produces the diffeomorphism anomaly and the second one is anomaly and contains the globally defined zero modes of the Ward identity. The explicit expression of this action is given on the complex plane.